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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License.Your use of this material constitutes acceptance of that license and the conditions of use of materials on this site. Copyright 2006, The Johns Hopkins University and William Brieger. All rights reserved. Use of these materials permitted only in accordance with license rights granted. Materials provided AS IS; no representations or warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for obtaining permissions for use from third parties as needed. JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Describing Data: Part I John McGready Johns Hopkins University

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Lecture Topics Types of data Measures of central tendency for continuous data Sample versus population Visually describing continuous data Underlying shape of the true distribution of continuous data 3

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section A Types of Data

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5 Steps in a Research Project 1.Planning 2.Design 3.Data collection 4.Data analysis 5.Presentation 6.Interpretation

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Continued 6 Biostatistics Issues Design of studies – Sample size – Role of randomization

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Continued 7 Biostatistics Issues Variability – Important patterns in data are obscured by variability – Distinguish real patterns from random variation

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8 Biostatistics Issues Inference – Draw general conclusions from limited data – For example: Survey Summarize

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Continued 9 1954 Salk Polio Vaccine Trial School Children Randomized Vaccinated N=200, 745 Placebo N=201, 229 Polio Cases:Vaccine 82 Placebo162

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10 1954 Salk Polio Vaccine Trial Reference: Meier, P. (1972), The Biggest Public Health Experiment Ever: The 1954 Field Trial of the Salk Poliomyelitis Vaccine, In: Statistics: A Guide to the Unknown, J. Tanur (Editor) Holden-Day.

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Continued 11 Design Features of the Polio Trial Comparison group Randomized Placebo controls Double blind Objective the groups should be equivalent except for the factor (vaccine) being investigated

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12 Design Features of the Polio Trial Question –Could the results be due to chance?

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13 Imbalance There were almost twice as many polio cases in the placebo compared to the vaccine group

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14 Could We Get Such Great Imbalance by Chance? Polio cases – Vaccine82 – Placebo162 – p-value = ? Statistical methods tell us how to make these probability calculations

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Continued 15 Types of Data Binary (dichotomous) data – Yes/No – Polio Yes/No – Cure Yes/No –GenderMale/Female

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Continued 16 Types of Data Categorical data (place individuals in categories) – Race/ethnicity nominal – Country of birth (no ordering) – Degree of agreement ordinal (ordering)

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Continued 17 Types of Data Continuous data (finer measurements) – Blood pressure – Weight – Height – Age

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18 Types of Data There are different statistical methods for different types of data

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19 Example: Binary Data To compare the number of polio cases in the two treatment arms of the Salk Polio vaccine, you could use... – Fishers Exact Test – Chi-Square test

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20 Example: Continuous Data To compare blood pressures in a clinical trial evaluating two blood pressure-lowering medications, you could use... – 2-sample T-test – Wilcoxon Rank Sum Test (nonparametric)

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section A Practice Problems

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22 Data Types State the data type of each of the following: Homicide rate (deaths/100,000) High school graduate (Y/N) Hair color Hospital expenditures (yearly, in dollars) Smoking status (none, light, heavy) Coronary heart disease (Y/N)

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section A Practice Problem Solutions

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Continued 24 Solutions Homicide rate (deaths/100,000)

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Continued 25 Solutions Homicide rate (deaths/100,000) Continuous

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Continued 26 Solutions High school graduate (Y/N)

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Continued 27 High school graduate (Y/N) Solutions Dichotomous (Binary)

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Continued 28 Hair color Solutions

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Continued 29 Hair color Solutions Categorical (Nominal)

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Continued 30 Hospital expenditures (yearly, in dollars) Solutions

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Continued 31 Hospital expenditures (yearly, in dollars) Solutions Continuous

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Continued 32 Smoking status (none, light, heavy) Solutions

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Continued 33 Smoking status (none, light, heavy) Solutions Categorical (Ordinal)

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Continued 34 Coronary heart disease (Y/N) Solutions

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35 Coronary heart disease (Y/N) Solutions Dichotomous (Binary)

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Section B Measures of Central Tendency Sample Versus Population JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH

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37 Summarizing and Describing Continuous Data Measures of the center of data – Mean – Median

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38 Sample Mean X The Average or Arithmetic Mean Add up data, then divide by sample size ( n ) The sample size n is the number of observations (pieces of data)

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Continued 39 Example n = 5 Systolic blood pressures (mmHg) X 1 = 120 X 2 = 80 X 3 = 90 X 4 = 110 X 5 = 95

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40 Example X = 120 + 8090 5 + 110 + 95 = 99mmH g +

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41 Notes on Sample Mean X Formula n XiXi X = i=1 n

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42 Summation Sign In the formula to find the mean, we use the summation sign This is just mathematical shorthand for add up all of the observations n i = 1 XiXi = X1X1 + X2X2 + X3X3 +......+ XnXn

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Notes on Sample Mean Also called sample average or arithmetic mean Sensitive to extreme values – One data point could make a great change in sample mean Why is it called the sample mean? – To distinguish it from population mean 43

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Continued 44 Population Versus Sample Population The entire group you want information about – For example: The blood pressure of all 18- year-old male college students in the United States

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Continued 45 Population Versus Sample Sample A part of the population from which we actually collect information and draw conclusions about the whole population – For example: Sample of blood pressures N=five 18-year-old male college students in the United States

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Continued 46 Population Versus Sample The sample mean X is not the population mean µ

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Population Versus Sample Continued 47 Population Sample Population mean µ Sample mean X

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48 Population Versus Sample We don t know the population mean µ but would like to know it We draw a sample from the population We calculate the sample mean X How close is X to µ ? Statistical theory will tell us how close X is to µ

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49 Statistical Inference Statistical inference is the process of trying to draw conclusions about the population from the sample

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Continued 50 Sample Median The median is the middle number 809095110120

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Continued 51 Sample Median The sample median is not sensitive to extreme values – For example: If 120 became 200, the median would remain the same, but the mean would change to 115 809095110200

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52 Sample Median If the sample size is an even number 809095110120125 Median 95 + 110 = 102.5mmHg 2

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section B Practice Problems

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Continued 54 Practice Problems The following data is the annual income (in 1000s of dollars) taken from nine students in the Hopkins internet-based MPH program: 37 102 34 12 111 56 72 17 33

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55 Practice Problems 1. Calculate the sample mean income 2. Calculate the sample median income 3. What population could this sample represent? 4. Which would change by a larger amount the mean or median if the 34 were replaced by 17, and the 12 replaced by a 31?

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section B Practice Problem Solutions

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Continued 57 Solutions 1. Calculate the sample mean income Remember: Where n=9 and x 1 through x 9 represent the nine observed values of income x = n x I i =1 n

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Continued 58 Solutions x = 37 +102 + 34 +12 +111+56+72+17+33 9 x = 474 = 9 52.67 – The mean income in our sample is $52,670

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Continued 59 Solutions 2. Calculate the sample median income – To calculate the sample median, we must first order our data from the lowest value to the highest value

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Continued 60 Solutions – The ordered data set appears below 12 17 33 34 37 56 72 102 111

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61 Solutions – Because we have nine values (an odd number), we can just pick the middle value to calculate the median 12 17 33 34 37 56 72 102 111 Median = 37

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62 Solutions 3. What population could this sample represent? – It could be representative off all Johns Hopkins Internet-based MPH students; it would need to be made clear that this is a random sample in order for it to be called representative

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63 Solutions 4. Which would change by a larger amount the mean or the median if the 34 were replaced by 17, and the 12 replaced by a 31? – Notice that both changes do nothing to change the position of the median; therefore, the only statistic that would change is the mean

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section C Visually Displaying Continuous Data: Histograms; The Underlying Population Distribution

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65 Pictures of Data Continuous Variables Histograms – Means and medians do not tell whole story – Differences in spread (variability) – Differences in shape of the distribution

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Continued 66 How to Make a Histogram Consider the following data collected from the 1995 Statistical Abstracts of the United States – For each of the 50 United States, the proportion of individuals over 65 years of age has been recorded

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Continued 67 How to Make a Histogram State % State % AL 12.9 IN 12.8 NE 14.1 SC 11.9 AK 4.6IA 15.4 NV 11.3 SD 14.7 AZ 13.4KS 13.9 NH 11.9 TN 12.7 AR 14.8 KY 12.7 NJ 13.2 TX 10.2 CA 10.6 LA 11.4 NM 11.0 UT 8.8 CO 10.1ME 13.9 NY 13.2 VT 12.1 CT 14.2 MD 11.2 NC 12.5 VA 11.1 DE 12.7 MA 14.1 ND 14.7 WA 11.6 FL 18.4MI12.4 OH 13.4 WV 15.4 GA 10.1 MN 12.5 OK 13.6 WI 13.4 HI 12.1MI12.5 OR 13.7 WY 11.1 ID 11.6MO14.1 PA 15.9 IL 12.6 MT 13.3 RI 15.6 Source: Statistical Abstracts of the United States, 1995

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Continued 68 How to Make a Histogram State % State % AL 12.9 IN 12.8 NE 14.1 SC 11.9 AK 4.6IA 15.4 NV 11.3 SD 14.7 AZ 13.4KS 13.9 NH 11.9 TN 12.7 AR 14.8 KY 12.7 NJ 13.2 TX 10.2 CA 10.6 LA 11.4 NM 11.0 UT 8.8 CO 10.1ME 13.9 NY 13.2 VT 12.1 CT 14.2 MD 11.2 NC 12.5 VA 11.1 DE 12.7 MA 14.1 ND 14.7 WA 11.6 FL 18.4MI12.4 OH 13.4 WV 15.4 GA 10.1 MN 12.5 OK 13.6 WI 13.4 HI 12.1MI12.5 OR 13.7 WY 11.1 ID 11.6MO14.1 PA 15.9 IL 12.6 MT 13.3 RI 15.6 Source: Statistical Abstracts of the United States, 1995

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Continued 69 How to Make a Histogram State % State % AL 12.9 IN 12.8 NE 14.1 SC 11.9 AK 4.6IA 15.4 NV 11.3 SD 14.7 AZ 13.4KS 13.9 NH 11.9 TN 12.7 AR 14.8 KY 12.7 NJ 13.2 TX 10.2 CA 10.6 LA 11.4 NM 11.0 UT 8.8 CO 10.1ME 13.9 NY 13.2 VT 12.1 CT 14.2 MD 11.2 NC 12.5 VA 11.1 DE 12.7 MA 14.1 ND 14.7 WA 11.6 FL 18.4MI12.4 OH 13.4 WV 15.4 GA 10.1 MN 12.5 OK 13.6 WI 13.4 HI 12.1MI12.5 OR 13.7 WY 11.1 ID 11.6MO14.1 PA 15.9 IL 12.6 MT 13.3 RI 15.6 Source: Statistical Abstracts of the United States, 1995

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Continued 70 How to Make a Histogram Count the number of observations in each class Here are the observations: Class Count Class Count Class Count 4.0–5.01 9.0–10.0 014.0–15.07 5.0–6.0 010.0–11.0515.0–16.04 6.0–7.0 011.0–12.0916.0–17.00 7.0–8.0 012.0–13.01217.0–18.00 8.0–9.0 113.0–14.01018.0–19.01

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71 How to Make a Histogram Divide range of data into intervals (bins) of equal width Count the number of observations in each class Draw the histogram Label scales

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Continued 72 Histogram A histogram of the percent of residents older than 65 years in the 50 United States.

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Continued 73 Histogram A histogram of the percent of residents older than 65 years in the 50 United States.

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74 Histogram A histogram of the percent of residents older than 65 years in the 50 United States.

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Continued 75 Pictures of Data: Histograms Recall the blood pressure data on a sample of 113 men

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Continued 76 Pictures of Data: Histograms Histogram of the Systolic Blood Pressure for 113 men. Each bar spans a width of five mmHg on the horizontal axis. The height of each bar represents the number of individuals with SBP in that range.

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Continued 77 Pictures of Data: Histograms Histogram of the Systolic Blood Pressure for 113 men. Each bar spans a width of five mmHg on the horizontal axis. The height of each bar represents the number of individuals with SBP in that range.

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78 Pictures of Data: Histograms Yet another histogram of the same BP information on 113 men. Here, the bin width is one mmHg, perhaps giving more detail than is necessary.

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79 Other Examples Another way to present the data in a histogram is to label the y- axis with relative frequencies as opposed to counts. The height of each bar represents the percentage of individuals in the sample with BP in that range. The bar heights should add to one.

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80 Intervals How many intervals (bins) should you have in a histogram? – There is no perfect answer to this – Depends on sample size n – Rough rule of thumb: # Intervals n n Number of Intervals 10 About 3 50 About 7 100 About 10

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Continued 81 Shapes of the Distribution Three common shapes of frequency distributions Symmetrical and bell shaped Positively skewed or skewed to the right Negatively skewed or skewed to the left A B C

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82 Shapes of the Distribution Three less common shapes of frequency distributions A B C Bimodal Reverse J-shaped Uniform

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83 Distribution Characteristics Mode Peak(s) Median Equal areas point Mean Balancing point Mod e Median Mea n

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Continued 84 Distribution Characteristics Right skewed (positively skewed) – Long right tail – Mean > Median Mod e Median Mea n

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Continued 85 Shapes of Distributions Left skewed (negatively skewed) – Long left tail – Mean < Median Mea n Media n Mod e

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Continued 86 Shapes of Distributions Symmetric (right and left sides are mirror images) – Left tail looks like right tail – Mean = Median = Mode Mea n Median Mode

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Continued 87 Shapes of Distributions Outlier – An individual observation that falls outside the overall pattern of the graph

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88 Shapes of Distributions Outlier Florid a Alaska

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Continued 89 The Histogram and the Probability Density The same histogram of BP measurements from a sample of 113 men. We are going to compare this to a histogram for a larger sample, and for the entire population.

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Continued 90 The Histogram and the Probability Density The same histogram of BP measurements from a sample of 113 men. We are going to compare this to a histogram for a larger sample, and for the entire population.

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Continued 91 The Histogram and the Probability Density The Probability Density for BP values in the entire population of men because the population is infinite, there are no bars, and the y-axis can not have actual counts.

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92 The Histogram and the Probability Density The probability density is a smooth idealized curve that shows the shape of the distribution in the population Areas in an interval under the curve represent the percent of the population in the interval

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section C Practice Problems

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Continued 94 Practice Problems What kind of shape do you think the distribution for the following data would have? Hospital length of stay for 1,000 randomly selected patients at Johns Hopkins Blood pressure in all women

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95 Practice Problems Annual income for all JHU Internet-based MPH students (refer to results of last Practice Problems, assume a random sample) Number of hours of sporting events watched last week by a sample of 350 men and 350 women

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section C Practice Problem Solutions

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Continued 97 Solutions Hospital length of stay for 1,000 randomly selected patients at Johns Hopkins

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Continued 98 Solutions Hospital length of stay for 1,000 randomly selected patients at Johns Hopkins

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Continued 99 Solutions Blood pressure in all women

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Continued 100 Solutions Blood pressure in all women

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Continued 101 Solutions Annual income for all JHU Internet-based MPH students

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Continued 102 Solutions Annual income for all JHU Internet-based MPH students Median Mea n

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Continued 103 Solutions Number of hours of sporting events watched last week by a sample of 350 women and 350 men

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Continued 104 Solutions Number of hours of sporting events watched last week by a sample of 350 women and 350 men

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Continued 105 Solutions Number of hours of sporting events watched last week by a sample of 350 women and 350 men Mode for Women? Mode for Men?

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JOHNS HOPKINS BLOOMBERG SCHOOL of PUBLIC HEALTH Section D Stem and Leaf Plots, Box Plots

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Continued 107 Sample 113 Men Suppose we took another look at our random sample of 113 men and their blood pressure measurements One tool for visualizing the data is the histogram

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Continued 108 Histogram: BP for 113 males

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109 Sample 113 Men: Stem and Leaf Another common tool for visually displaying continuous data is the stem and leaf plot Very similar to a histogram – Like a histogram on its side – Allows for easier identification of individual values in the sample

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Continued 110 Stem and Leaf: BP for 113 Males

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Continued 111 Stem and Leaf: BP for 113 Males Stems

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Continued 112 Stem and Leaf: BP for 113 Males Leaves

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Continued 113 Stem and Leaf: BP for 113 Males

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Continued 114 Stem and Leaf: BP for 113 Males

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Continued 115 Sample 113 Men: Stem and Boxplot Another common visual display tool is the boxplot – Gives good insight into distribution shape in terms of skewness and outlying values – Very nice tool for easily comparing distribution of continuous data in multiple groups can be plotted side by side

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Continued 116 Boxplot: BP for 113 Males Boxplot of Systolic Blood Pressures Sample of 113 Men

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Continued 117 Boxplot: BP for 113 Males Sample Median Blood Pressure Boxplot of Systolic Blood Pressures Sample of 113 Men

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Continued 118 Boxplot: BP for 113 Males 75 th Percentile 25 th Percentile Boxplot of Systolic Blood Pressures Sample of 113 Men

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Continued 119 Boxplot: BP for 113 Males Largest Observation Smallest Observation Boxplot of Systolic Blood Pressures Sample of 113 Men

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120 Hospital Length of Stay for 1,000 Patients Suppose we took a sample of discharge records from 1,000 patients discharged from a large teaching hospital How could we visualize this data?

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121 Histogram: Length of Stay Length of Stay (Days) Frequency

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Continued 122 Boxplot: Length of Stay Ho spital LO S (D ays) fo r 1,000 Patients

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Continued 123 Boxplot: Length of Stay Ho spital LO S (D ays) fo r 1,000 Patients

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Continued 124 Boxplot: Length of Stay Largest Non-Outlier Smallest Non-Outlier Ho spital LO S (D ays) fo r 1,000 Patients

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Continued 125 Boxplot: Length of Stay Large Outliers Ho spital LO S (D ays) fo r 1,000 Patients

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